Tangentialflächenvektors

Tangentialfläc is a term from a fictional branch of differential geometry that describes a structured collection of tangential facets attached to a smooth surface along a parameterized curve. The concept is used in theoretical discussions and speculative writings to illustrate how local tangential information can be organized into a mesh-like representation without invoking full curvature data.

Definition and construction: Let S be a C^2 surface in three-dimensional space and γ: [a,b] → S a smooth curve on S. For each t in [a,b], select a finite set of directions within the tangent plane T_{γ(t)}S. The tangential facets at γ(t) are the convex hulls of the corresponding tangent vectors projected into R^3, forming a piecewise-linear patch that approximates the neighborhood of γ(t). The Tangentialfläc is the union of these patches along t, with a chosen sampling rate. In the limit of dense sampling and refined directional resolution, the Tangentialfläc approaches a continuous tangential field along γ.

Properties and uses: The Tangentialfläc emphasizes tangential information without requiring full second-order data. It can be robust to moderate noise and is compatible with mesh generation, visualization, and terrain modeling in fictional or educational contexts. It differs from standard tangent-plane or normal-map techniques by emphasizing a finite, directional, facet-based approximation.

History and reception: The term Tangentialfläc originated in a narrative or pedagogical context and is not part of conventional mathematical practice. It appears in fictional texts to explore how geometry, visualization, and computation intersect. Related concepts include the tangent plane, the tangent cone, and polyhedral meshing.